Scaling Laws from Sequential Feature Recovery: A Solvable Hierarchical Model

TL;DR

This paper introduces a hierarchical model demonstrating how scaling laws naturally emerge from feature learning in multi-layer networks, with sequential feature recovery and explicit error decay.

Contribution

It presents a simple mechanism and spectral algorithm that achieve improved scaling and recover features sequentially in a hierarchical setting.

Findings

Spectral algorithm improves scaling over shallow methods.

Strong features are detected at small sample sizes.

Prediction error decays following an explicit power law.

Abstract

We propose a simple mechanism by which scaling laws emerge from feature learning in multi-layer networks. We study a high-dimensional hierarchical target that is a globally high-degree function, but that can be represented by a combination of latent compositional features whose weights decrease as a power law. We show that a layer-wise spectral algorithm adapted to this compositional structure achieves improved scaling relative to shallow, non-adaptive methods, and recovers the latent directions sequentially: strong features become detectable at small sample sizes, while weaker features require more data. We prove sharp feature-wise recovery thresholds and show that aggregating these transitions yields an explicit power-law decay of the prediction error. Technically, the analysis relies on random matrix methods and a resolvent-based perturbation argument, which gives matching upper and lower bounds for individual eigenvector recovery beyond what standard gap-based perturbation bounds provide. Numerical experiments confirm the predicted sequential recovery, finite-size smoothing of the thresholds, and separation from non-hierarchical kernel baselines. Together, these results show how smooth scaling laws can emerge from a cascade of sharp feature-learning transitions.